Trilateration is a common technique for finding a unique point for the given three circles. Extending this fact to 3d, I would like to know the minimum number of spheres required to compute a point uniquely in 3d and the algorithm for finding the same
Thanks in advance.
Motivation:I've recently started taking part in Kaggle competitions. As a part of Google smartphone decimeter challenge, I wanted to compute the loction using this method.
In two dimensions, knowing the distance of a point $X$ to two fixed points $A$ and $B$ will give you two possible positions of $X$ (unless $X$ lies on the line $AB$). If you don't know which side of the line $X$ lies on, you need a third measurement from a point $C$ not on the line $AB$.
In three dimensions, knowing the distance of a point $X$ to two fixed points $A$ and $B$ gives you a circle in $\Bbb R^3$, whose plane is perpendicular to $AB$; and you need a third measurement from a point $C$ not on $AB$ to narrow it down to two possibilities. These two possibilities are reflections of each other in the plane containing $ABC$. As for two dimensions, if you don't know which side of the plane $X$ lies on, you need a fourth measurement from a point $D$ not in the plane containing $ABC$.